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Euler Lagrange Equation Question

  1. Apr 1, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider the function f(y,y',x) = 2yy' + 3x2y where y(x) = 3x4 - 2x +1. Compute ∂f/∂x and df/dx. Write both solutions of the variable x only.


    2. Relevant equations

    Euler Equation: ∂f/∂y - d/dx * ∂f/∂y' = 0



    3. The attempt at a solution

    Would I first just find ∂f/∂y and ∂f/∂y' as follows:

    ∂f/∂y = 2y' + 3x2
    ∂f/∂y' = 2y

    and then insert y(x) and y'(x) into my two equations for ∂f/∂y and ∂f/∂y':

    y(x) = 3x4 - 2x +1
    y'(x) = 12x3 - 2

    → ∂f/∂y = 2(12x3 - 2) + 3x2
    → ∂f/∂y' = 2(3x4 - 2x +1)

    This is where I begin to get lost. Would I plug these values into my Euler equation to find an expression for df/dx and ∂f/∂x?

    I would attempt this but I want to see if I'm headed in the right direction or not first. Any help would be much appreciated.
     
  2. jcsd
  3. Apr 2, 2013 #2

    BruceW

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    Homework Helper

    You don't need to find those at all. You don't need to use the Euler-Lagrange equations either. Just calculate the things they ask for in the question. I think they are just testing your knowledge of how to do partial differentiation and the full derivative.
     
  4. Apr 2, 2013 #3
    Thanks for you response. I feel as if I made this problem more complicated than it should.

    For ∂f/∂x:

    ∂f/∂x = 6xy = 6x(3x4 - 2x +1)

    and df/dx:

    df/dx = ∂f/∂y * dy/dx
    = d/dy(3x2y + 2yy') * (12x3 - 2)
    = (3x2 + 2(d/dy(yy'))) * (12x3 - 2)
    = (3x2 + 2(y * d/dy(y') + y')) * (12x3 - 2)

    However I'm a bit confused on how to take the derivative of d/dy(y')
    Would I just treat y' as a fraction and cancel out dy's like so: y * d/dy(dy/dx) = d/dx * y = y'
     
  5. Apr 2, 2013 #4

    BruceW

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    Homework Helper

    I think your answer for ∂f/∂x is correct. But I don't agree with your answer for df/dx. The line: df/dx = ∂f/∂y * dy/dx I think is not right. This is not the correct expression for the full derivative. p.s. you can realise this by thinking what is being held constant for ∂f/∂y
     
    Last edited: Apr 2, 2013
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